The re-iterated Galerkin method

The iterated Galerkin method introduced by Sloan is modifiedto allow the improvement in the Galerkin approximation whichiteration produces to be repeated. It is shown in principle,and verified in illustration, that the repeated improvementsmay be achieved under weaker conditions than those requiredby Sloan. The sequence of approximations obtained is appliedto variational principles and in the case when a maximum principleis available it is shown that each step in the process improvesthe estimate of the maximum value of the associated functional.Applications to integral equations are given.